smoked one to nine cigarettes a day, and 473 had mothers who had smoked 10 or more cigarettes a day. The figure shows the 95 per cent confidence intervals for differences in mean weight according to sex of baby and smoking habits of mothers: at birth, and at three and six months.

The results show, for example, that at birth, the difference between the sample mean weight of female babies born to non-smoking mothers and those born to mothers smoking 10 or more cigarettes a day, was (3220 − 3052) = 168 g. That is, the infants of smoking mothers are on average lighter by 168 g. Is this difference statistically significant in the population, or due simply to chance? The 95 per cent confidence interval of (−234 to −102) g, does not include zero, so you can be 95 per cent confident that the difference is real, i.e. is statistically significant.

Exercise 10.1 Interpret the sample mean and confidence intervals shown in Table 10.2 for all four differences in weights at six months.

Estimating the difference between two matched population means – using a method based on the matched-pairs t test

If the data within each of the two groups whose means you are comparing is widely spread compared to the difference in the spreads between the groups,5 this can make it more difficult to detect any difference in their means. When data is matched (see Chapter 7 for an explanation of matching), this reduces much of the within-group variation, and, for a given sample size, makes it easier to detect any differences between groups. As a consequence, you can achieve better precision (narrower confidence intervals), without having to increase sample size. The disadvantage of matching is that it is sometimes difficult to find a sufficiently large number of matches (as you saw in the case-control discussion earlier).

In the independent groups case, the mean of each group is computed separately, and then a confidence interval for the difference in these means is calculated. In the matched groups case, we use a method based on the matched-pairs t test, in which the difference between each pair of values is computed first and then a confidence interval for the mean of these differences is calculated.

An example from practice

Table 10.3 shows the 95 per cent confidence intervals for the difference in bone mineral density in two matched groups of women, one group depressed and one ‘normal’ (Michelson et al. 1995). (Ignore the ‘SD from expected peak’ rows.) Only one of the confidence intervals contains zero, indicating that there is no difference in population mean bone mineral density at the radius, but there is at all of the other five sites.

5 Called ‘between-group’ variation.

W = 790.5

Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0668 The test is significant at 0.0616 (adjusted for ties)

Cannot reject at alpha = 0.05

Figure 10.3 Minitab’s Mann–Whitney output for a 95 per cent confidence interval for the difference between two independent median Apgar scores – for infants born in maternity units and at home (raw data in Table 10.1). Note that Minitab uses Greek ‘ETA’ to denote the population median

similar. However, if the rank sums are different, you need to know whether this difference could simply be due to chance, or is because there really is a statistically significant difference in the population medians. A Mann–Whitney confidence interval for the difference will help you decide between these alternatives.

As an illustration, let’s compare the difference in the population median Apgar scores for the maternity unit and home birth infants, using the sample data in Table 10.1. These are independent groups, but since this data is ordinal, we cannot use the two-sample t test, but we can use the Mann–Whitney test of medians. The output from Minitab is shown in Figure 10.3, with the 95 per cent confidence interval in the fourth row.7 Since the confidence interval of (−2 to 0) contains zero, you must conclude that the difference in the population median Apgar scores is not statistically significant. Notice that the confidence level is given as 95.2 per cent, not 95 per cent. Confidence intervals for medians cannot always achieve the precise confidence level you asked for, because of the way in which a median is calculated.

An example from practice

Table 10.4 is from a randomised controlled double-blind trial to compare the cost effectiveness of two treatments in relieving pain after blunt instrument injury in an A&E department (Rainer et al. 2000). It shows the median times spent by two groups of patients in various clinical situations. One group received ketorolac, the other group morphine. The penultimate column contains the 95 per cent confidence intervals for the difference in various median treatment times (minutes), between the groups (ignore the last column). As the footnote to the table indicates, these results were obtained using the Mann–Whitney method.

The only confidence interval not containing zero is that for the difference in median ‘time between receiving analgesia and leaving A&E’, for which the difference in the sample medians is 20.0 minutes. So this is the only treatment time for which the difference in population median

7 As far as I am aware, SPSS does not appear to calculate a confidence interval for two independent medians.